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The Charm of Banach's Theorem: How to Find the Precise Fixed Point?
Fixed point calculation is the process of finding exact or approximate fixed points of a given function. In its most common form, a given function satisfies the conditions of Brouwer's fixed-point theorem: that is, the function is continuous and maps unit d-cubes onto itself. Brouwer's fixed point theorem guarantees that the function has a fixed point, but its proof is not constructive.
This has led to the creation of various algorithms designed to calculate approximate fixed points and are widely used in economics, game theory and dynamic system analysis.
Before discussing fixed points, it is necessary to understand some basic definitions. The unit interval is denoted E := [0, 1], and the unit d-dimensional cube is denoted E^d. A continuous function f defined on E^d is a map from E^d to itself. It is often assumed that this function is not only continuous, but also Lipschitz continuous, that is, there is a constant L such that for all x and y, |f(x) - f(y)| ≤ L ⋅ |x - y |.
A fixed point x is a point in E^d such that f(x) = x. According to Brouwer's fixed point theorem, any continuous function has a fixed point from E^d to itself.
Although for general functions it is impossible to calculate the fixed point exactly because it can be any real number, the fixed point calculation algorithm seeks to approximate the fixed point. The usual standards are as follows:
Residual criterion: Given an approximate parameter ε > 0, an ε-residual fixed point is defined as a point x such that |f(x) - x| ≤ ε.
Absolute criterion: For a given parameter δ > 0, a δ-absolute fixed point is a point x such that |x - x₀| ≤ δ, where x₀ is any fixed point.
Relative standard: The condition is |x - x₀|/|x₀| ≤ δ, x₀ satisfies f(x₀) = x₀.
For Lipschitz continuous functions, the absolute criterion is stronger than the residual criterion. This becomes particularly important if f is a Lipschitz continuous function that satisfies the definition.
The most basic step of the fixed-point calculation algorithm is value query. Given any x in E^d, the algorithm provides the value f(x) of the function f by an oracle. The accuracy of the approximate fixed point depends on the oracle accuracy. However, for these different calculation methods there are many types based on Lipschitz continuity, including algorithms derived from the famous Banach fixed point theorem.
Of course, for contraction functions, the calculation of fixed points is obviously much simpler. According to Banach's fixed point theorem, every contraction function that satisfies Brouwer's condition has a unique fixed point. The fixed-point iteration algorithm is one of the earliest algorithms. The error after t iterations decreases exponentially, so the number of iterations typically required for a delta-relative fixed point in d-dimensional space can be expressed as a logarithmic ratio.
When d increases, Banach's algorithm clearly shows its superiority, especially in terms of computational complexity at fixed points, and provides a convenient solution for solving problems in high-dimensional space.
In the case of differentiable functions, Newton's method can often significantly speed up calculations if the algorithm can evaluate its derivatives. However, for general functions with Lipschitz constant greater than 1, the difficulty of computing the fixed point increases significantly, which involves an infinite number of evaluation queries and becomes a thorny challenge.
Although the calculation of one-dimensional functions is relatively simple, for two-dimensional and higher-dimensional functions, the finding and calculation of fixed points becomes extremely challenging. Nowadays, many methods based on function evaluation have been proposed. For example, the algorithm developed by Herbert Scarfe in 1967 is one of them. By forming a fully labeled "original set", ε-residual fixation is achieved point approximation.
With in-depth research on fixed-point calculations, the complexity of related algorithms and corresponding inspirations are becoming increasingly abundant. With applications in different fields, how to find these fixed points more efficiently and accurately remains a major challenge in mathematics and computer science.
While exploring these mathematical mysteries, we can’t help but ask: In real life, can we also apply similar mathematical principles to find fixed points to solve problems?
